Question 2.21: (Expectation of an Exponential Random Variable) Let X be exp......
(Expectation of an Exponential Random Variable) Let X be exponentially distributed with parameter λ. Calculate E[X].
E[X]=\int_{0}^{\infty}x\lambda e^{-\lambda x}\,d x
Integrating by parts (d v=\lambda e^{-\lambda x},u=x) yields
E[X]=-x e^{-\lambda x}{\big|}_{0}^{\infty}+\int_{0}^{\infty}e^{-\lambda x}\,d x=0-{\frac{e^{-\lambda x}}{\lambda}}{\bigg|}_{0}^{\infty}
={\frac{1}{\lambda}}
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